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First derivative of the legendre polynomials

  1. Jan 14, 2014 #1
    show that the first derivative of the legendre polynomials satisfy a self-adjoint differential equation with eigenvalue λ=n(n+1)-2


    The attempt at a solution:

    (1-x^2 ) P_n^''-2xP_n^'=λP_n
    λ = n(n + 1) - 2 and (1-x^2 ) P_n^''-2xP_n^'=nP_(n-1)^'-nP_n-nxP_n^'
    ∴nP_(n-1)^'-nP_n-nxP_n^'=( n(n + 1) - 2 )P_n
    For n=2
    2P_1^'-2P_2-2xP_2^'=4P_n
    Using legendre polynomial properties we have
    -9x^2+2x+1=2(3x^2-1)
    ∴-15x^2+2x+3=0
    I don’t know what to do from here.
     
  2. jcsd
  3. Jan 15, 2014 #2

    HallsofIvy

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    Did you consider just replacing the first derivative with just "y" so that the equation becomes
    [tex](1- x^2)y'- 2xy= \lambda P_n[/tex]
    Differentiating
    [tex](1- x^2)y''- 4xy'- 2y= \lambda P_n'= \lambda y[/tex]

    [tex](1- x^2)y''- 4xy'= (2+ \lambda)y[/tex]
     
  4. Jan 15, 2014 #3
    Thanks Hallsofvy
     
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